How to represent an expression as a monomial. Reducing a monomial to standard form, examples, solutions

Monomial is an expression that is the product of two or more factors, each of which is a number expressed by a letter, digits or power (with a non-negative integer exponent):

2a, a 3 x, 4abc, -7x

Since the product of identical factors can be written as a power, a single power (with a non-negative integer exponent) is also a monomial:

(-4) 3 , x 5 ,

Since a number (integer or fraction), expressed by a letter or numbers, can be written as the product of this number by one, any individual number can also be considered as a monomial:

x, 16, -a,

Standard form of monomial

Standard form of monomial is a monomial that has only one numerical factor, which must be written in first place. All variables are in alphabetical order and are contained in a monomial only once.

Numbers, variables and powers of variables also belong to monomials of the standard form:

7, b, x 3 , -5b 3 z 2 - monomials of standard form.

The numerical factor of a monomial of standard form is called coefficient of the monomial. Monomial coefficients equal to 1 and -1 are usually not written.

If a monomial of standard form does not have a numerical factor, then it is assumed that the coefficient of the monomial is equal to 1:

x 3 = 1 x 3

If a monomial of standard form does not have a numerical factor and is preceded by a minus sign, then it is assumed that the coefficient of the monomial is equal to -1:

-x 3 = -1 · x 3

Reducing a monomial to standard form

To bring a monomial to standard form you need to:

1. Multiply numerical factors if there are several of them. Raise a numeric factor to a power if it has an exponent. Put the numeric factor first.
2. Multiply all the same variables so that each variable appears only once in the monomial.
3. Arrange the variables after the numeric factor in alphabetical order.

Example. Present the monomial in standard form:

a) 3 yx 2 (-2) y 5 x; b) 6 bc · 0.5 3

ab

a) 3 yx 2 (-2) y 5 x Solution: x 2 xyy 5 = -6x 3 y 6
= 3 (-2) b) 6 bc · 0.5 b) 6 · 0.5b 3 3 = 6 0.5 = 3· 0.5 4 3 = 6 0.5

c

Power of a monomial Power of a monomial

is the sum of the exponents of all letters included in it.

If a monomial is a number, that is, it does not contain variables, then its degree is considered equal to zero. For example:

Therefore, to find the degree of a monomial, you need to determine the exponent of each of the letters included in it and add these exponents. If the exponent of a letter is not specified, then it is equal to one.

Examples:

So how are u x the exponent is not specified, which means it is equal to 1. The monomial does not contain other variables, which means its degree is equal to 1.

A monomial contains only one variable to the second power, which means the degree of this monomial is 2.

3) · 0.5 3 3 = 6 0.5 2 d

Index a equals 1, exponent b- 3, indicator 3 = 6 0.5- 2, indicator d- 1. The degree of this monomial is equal to the sum of these indicators.

I. Expressions that are made up of numbers, variables and their powers using the action of multiplication are called monomials.

Examples of monomials:

A) a; b) ab; V) 12; G)-3c; d) 2a 2 ∙(-3.5b) 3 ; e)-123.45xy 5 z; and) 8ac∙2.5a 2 ∙(-3c 3).

II. This type of monomial, when the numerical factor (coefficient) comes first, followed by the variables with their powers, is called the standard type of monomial.

Thus, the monomials given above, under the letters a B C), G) And e) written in standard form, and the monomials under the letters d) And and) it is required to bring it to a standard form, i.e. to a form where the numerical factor comes first, followed by the letter factors with their exponents, and the letter factors are in alphabetical order. Let us present the monomials d) And and) to the standard view.

d) 2a 2 ∙(-3.5b) 3=2a 2 ∙(-3.5) 3 ∙b 3 =-2a 2 ∙3.5∙3.5∙3.5∙b 3 = -85.75a 2 b 3 ;

and) 8ac∙2.5a 2 ∙(-3c 3)=-8∙2.5∙3a 3 c 3 = -60a 3 c 3 .

III.The sum of the exponents of all variables included in a monomial is called the degree of the monomial.

Examples. What degree do monomials have? a) - g)?

a) a. First;

b) ab. Second: A in the first degree and b to the first power - the sum of indicators 1+1=2 ;

V) 12. Zero, since there are no letter factors;

G) -3c. First;

d) -85.75a 2 b 3 . Fifth. We have reduced this monomial to standard form, we have A to the second degree and b in the third. Let's add up the indicators: 2+3=5 ;

e) -123.45xy 5 z. Seventh. We added up the exponents of the letter factors: 1+5+1=7 ;

and) -60a 3 c 3 . Sixth, since the sum of the exponents of the letter factors 3+3=6 .

IV. Monomials that have the same letter part are called similar monomials.

Example. Indicate similar monomials among the given monomials 1) -7).

1) 3aabbc; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 2 bac; 5) 10aaa 2 x; 6) -2.3a 4 x; 7) 34x 2 y.

Let us present the monomials 1), 4) And 5) to the standard view. Then the line of monomials data will look like this:

1) 3a 2 b 2 c; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 3 bc; 5) 10a 4x; 6) -2.3a 4 x; 7) 34x 2 y.

Similar will be those that have the same letter part, i.e. 1) and 3) ; 2) and 4); 5) and 6).

1) 3a 2 b 2 c and 3) 56a 2 b 2 c;

2) -4.1a 3 bc and 4) 98.7a 3 bc;

5) 10a 4 x and 6) -2.3a 4 x.

We noted that any monomial can be bring to standard form. In this article we will understand what is called bringing a monomial to standard form, what actions allow this process to be carried out, and consider solutions to examples with detailed explanations.

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What does it mean to reduce a monomial to standard form?

It is convenient to work with monomials when they are written in standard form. However, quite often monomials are specified in a form different from the standard one. In these cases, you can always go from the original monomial to a monomial of the standard form by performing identity transformations. The process of carrying out such transformations is called reducing a monomial to a standard form.

Let us summarize the above arguments. Reduce the monomial to standard form- this means performing identical transformations with it so that it takes on a standard form.

How to bring a monomial to standard form?

It's time to figure out how to reduce monomials to standard form.

As is known from the definition, monomials of non-standard form are products of numbers, variables and their powers, and possibly repeating ones. And a monomial of the standard form can contain in its notation only one number and non-repeating variables or their powers. Now it remains to understand how to bring products of the first type to the type of the second?

To do this you need to use the following the rule for reducing a monomial to standard form consisting of two steps:

• First, a grouping of numerical factors is performed, as well as identical variables and their powers;
• Secondly, the product of the numbers is calculated and applied.

As a result of applying the stated rule, any monomial will be reduced to a standard form.

Examples, solutions

All that remains is to learn how to apply the rule from the previous paragraph when solving examples.

Example.

Reduce the monomial 3 x 2 x 2 to standard form.

Solution.

Let's group numerical factors and factors with variable x. After grouping, the original monomial will take the form (3·2)·(x·x 2) . The product of numbers in the first brackets is equal to 6, and the rule for multiplying powers with the same bases allows the expression in the second brackets to be represented as x 1 +2=x 3. As a result, we obtain a polynomial of the standard form 6 x 3.

Here is a short summary of the solution: 3 x 2 x 2 =(3 2) (x x 2)=6 x 3.

Answer:

3 x 2 x 2 =6 x 3.

So, to bring a monomial to a standard form, you need to be able to group factors, multiply numbers, and work with powers.

To consolidate the material, let's solve one more example.

Example.

Present the monomial in standard form and indicate its coefficient.

Solution.

The original monomial has a single numerical factor in its notation −1, let's move it to the beginning. After this, we will separately group the factors with the variable a, separately with the variable b, and there is nothing to group the variable m with, we will leave it as is, we have . After performing operations with powers in brackets, the monomial will take the standard form we need, from which we can see the coefficient of the monomial equal to −1. Minus one can be replaced with a minus sign: .

The concept of a monomial

Definition of a monomial: A monomial is an algebraic expression that uses only multiplication.

Standard form of monomial

What is the standard form of a monomial? A monomial is written in standard form, if it has a numerical factor in the first place and this factor is called the coefficient of the monomial, there is only one in the monomial, the letters of the monomial are arranged in alphabetical order and each letter appears only once.

An example of a monomial in standard form:

here in the first place is the number, the coefficient of the monomial, and this number is only one in our monomial, each letter occurs only once and the letters are arranged in alphabetical order, in this case it is the Latin alphabet.

Another example of a monomial in standard form:

each letter appears only once, they are arranged in Latin alphabetical order, but where is the coefficient of the monomial, i.e. the numeric factor that should come first? Here it is equal to one: 1adm.

Can the coefficient of a monomial be negative? Yes, maybe, example: -5a.

Can the coefficient of a monomial be fractional? Yes, maybe, example: 5.2a.

If a monomial consists only of a number, i.e. has no letters, how can I bring it to standard form? Any monomial that is a number is already in standard form, for example: the number 5 is a monomial in standard form.

Reducing monomials to standard form

How to bring a monomial to standard form? Let's look at examples.

Let the monomial 2a4b be given; we need to bring it to standard form. We multiply its two numerical factors and get 8ab. Now the monomial is written in standard form, i.e. has only one numerical factor, written in the first place, each letter in the monomial appears only once and these letters are arranged in alphabetical order. So 2a4b = 8ab.

Given: monomial 2a4a, bring the monomial to standard form. We multiply the numbers 2 and 4, replacing the product aa with the second power of a 2. We get: 8a 2 . This is the standard form of this monomial. So 2a4a = 8a 2 .

Similar monomials

What are similar monomials? If monomials differ only in coefficients or are equal, then they are called similar.

Example of similar monomials: 5a and 2a. These monomials differ only in coefficients, which means they are similar.

Are the monomials 5abc and 10cba similar? Let's bring the second monomial to standard form and get 10abc. Now we can see that the monomials 5abc and 10abc differ only in their coefficients, which means that they are similar.

Addition of monomials

What is the sum of the monomials? We can only sum similar monomials. Let's look at an example of adding monomials. What is the sum of monomials 5a and 2a? The sum of these monomials will be a monomial similar to them, the coefficient of which is equal to the sum of the coefficients of the terms. So, the sum of the monomials is 5a + 2a = 7a.

More examples of adding monomials:

2a 2 + 3a 2 = 5a 2
2a 2 b 3 c 4 + 3a 2 b 3 c 4 = 5a 2 b 3 c 4

Again. You can only add similar monomials; addition comes down to adding their coefficients.

Subtracting monomials

What is the difference between the monomials? We can only subtract similar monomials. Let's look at an example of subtracting monomials. What is the difference between monomials 5a and 2a? The difference of these monomials will be a monomial similar to them, the coefficient of which is equal to the difference of the coefficients of these monomials. So, the difference of the monomials is 5a - 2a = 3a.

More examples of subtracting monomials:

10a 2 - 3a 2 = 7a 2
5a 2 b 3 c 4 - 3a 2 b 3 c 4 = 2a 2 b 3 c 4

Multiplying monomials

What is the product of monomials? Let's look at an example:

those. the product of monomials is equal to a monomial whose factors are made up of the factors of the original monomials.

Another example:

2a 2 b 3 * a 5 b 9 = 2a 7 b 12 .

How did this result come about? Each factor contains “a” to the power: in the first - “a” to the power of 2, and in the second - “a” to the power of 5. This means that the product will contain “a” to the power of 7, because when multiplying identical letters, the exponents of their powers fold up:

A 2 * a 5 = a 7 .

The same applies to the factor “b”.

The coefficient of the first factor is two, and the second is one, so the result is 2 * 1 = 2.

This is how the result was calculated: 2a 7 b 12.

From these examples it is clear that the coefficients of monomials are multiplied, and identical letters are replaced by the sums of their powers in the product.